I wanted to prove the following bundle isomorphisms, on page 20 of hatcher's $K$-theroy. That is $$f^*(E_1) \otimes f^*(E_2) \cong f^*(E_1 \otimes E_2)$$
$$f^*(E_1) \oplus f^*(E_2) \cong f^*(E_1 \oplus E_2)$$ So I want to show they satisfy the UP. Indeed, there is a well defined map, for each $i =1,2$, $$ f^*(E_i) \xrightarrow{f'_i} E_i$$ Then fiberwise we have a linear isomorphism of (say for example) $$ f^*(E_1) \otimes f^*(E_1) \xrightarrow {f'_1 \otimes f'_2} E_1 \otimes E_2$$ This map solves the universal property of pull back, except that it is defined fiberwise so we haven't proved it is continuous. How do I prove this?
Thoughts: I think this follows from how we define the topology. From my previous question, we interpret, with $B$ as base space, $$E_1 \otimes E_2 := \bigsqcup_{x \in B} (E_1)_x \otimes (E_2)_x. $$ and a basis open set is of the form $$ \bigsqcup_{x \in U} (E_1)_x \otimes (E_2)_x$$ where $U$ is taken to be a common intersection of the coverings from spaces $E_i$. So the preimage of this set under $f'_1 \otimes f'_2$ is $$ \bigsqcup_{y \in f^{-1}(U)} (f^*(E_1)_y \otimes f^*(E_2)_y)$$ which is an open set. Hence, as we have shown this for basis elements, the full map is continuous.
Your thoughts concerning topologizing the bundles are correct. After the proof of Proposition 1.5 Hatcher gets explicit about local trivializations of the pullback bundle $f^*(E)$, and that is all you need. For each $x \in B$ choose an open neighborhood $U$ such that both $E_1, E_2$ (and hence also $E_1 \otimes E_2$) are trivial over $U$. This canonically induces trivializations of $f^*(E_1),f^*(E_2), f^*(E_1 \otimes E_2)$ over $f^{-1}(U)$. With respect to these trivializations the fiberwise isomorphism $f^*(E_1) \otimes f^*(E_1) \to E_1 \otimes E_2$ corresponds on $f^{-1}(U)$ to $f \times id : f^{-1}(U) \times (\mathbb{R}^{n_1} \otimes \mathbb{R}^{n_2}) \to U \times (\mathbb{R}^{n_1} \otimes \mathbb{R}^{n_2})$ which is obviously is continuous. Hence $f^*(E_1) \otimes f^*(E_1) \to E_1 \otimes E_2$ is locally continuous which suffices to prove the desired result.
Another way to see it is to consider the obvious fiberwise isomorphism $f^*(E_1) \otimes f^*(E_1) \to f^*(E_1 \otimes E_2)$ of bundles over $A$. On $f^{-1}(U)$ it corresponds to the identity on $f^{-1}(U) \times (\mathbb{R}^{n_1} \otimes \mathbb{R}^{n_2})$. Hence it is locally a bundle isomorphism which again suffices to prove the result.
By the way, perhaps you should consult also other books. Hatcher is a little short when dealing with the basic material. For example, he does not mention the relationship of bundles with the transition maps $U_\alpha \cap U_\beta \to GL_n(\mathbb{R})$ associated to a bundle atlas (see Andres Mejia's comment).
Here are some books:
Steenrod, Norman Earl. The topology of fibre bundles. Vol. 14. Princeton University Press, 1999
Husemoller, Dale. Fibre bundles. Vol. 5. New York: McGraw-Hill, 1966. https://www.maths.ed.ac.uk/~v1ranick/papers/husemoller
Cohen, Ralph. The Topology of Fiber Bundles. http://math.stanford.edu/~ralph/fiber.pdf
Atiyah, Michael. K-theory. CRC Press, 2018. https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf